Question

in the diagram below, \\( \triangle a b c \\) is equilateral and \\( \angle a c d \cong \angle b c d \\).
william wants to prove that \\( \overline{a d} \cong \overline{b d} \\). which of these statements can william make about the diagram to usein his proof? select all that apply.
\\( \triangle a c d \cong \triangle b c d \\) \\( \overline{a c} \cong \overline{b c} \\)
\\( \triangle a b c \sim \triangle a b d \\) \\( \overline{b c} \cong \overline{c d} \\)
\\( \overline{a c} \cong \overline{c d} \\) \\( \overline{c d} \cong \overline{c d} \\)

Explanation:

Step1: Analyze triangle \(ABC\)

Since \(\triangle ABC\) is equilateral, by the property of an equilateral triangle, all its sides are equal. So, \(\overline{AC}\cong\overline{BC}\).

Step2: Consider the common side

In \(\triangle ACD\) and \(\triangle BCD\), \(\overline{CD}\cong\overline{CD}\) (reflexive property of congruence).

Step3: Use the given angle congruence

Given \(\angle ACD\cong\angle BCD\).

Step4: Apply the Side - Angle - Side (SAS) congruence criterion

For \(\triangle ACD\) and \(\triangle BCD\), we have \(\overline{AC}\cong\overline{BC}\), \(\angle ACD\cong\angle BCD\), and \(\overline{CD}\cong\overline{CD}\). So, \(\triangle ACD\cong\triangle BCD\) by SAS.

Step5: Use the property of congruent triangles

If \(\triangle ACD\cong\triangle BCD\), then their corresponding sides \(\overline{AD}\cong\overline{BD}\) (corresponding parts of congruent triangles are congruent).

Answer:

\(\triangle ACD\cong\triangle BCD\), \(\overline{AC}\cong\overline{BC}\), \(\overline{CD}\cong\overline{CD}\)